You can't isolate vss. 35-40 or you can come to some wrong conclusions. Vss. 44-45 explains vss. 35-40.
I never said you had to in fact above I encouraged everyone to explain the passages in the context of John 6.
Jn 6:44 No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day.
45 It is written in the prophets, And they shall be all taught of God. Every man therefore that hath heard, and hath learned of the Father, cometh to me.
Vs. 45 explains the word "given" in vss. 37 and 39. If a person comes to Christ, it is because he has been taught by the Father. How was this done? Was it done through regeneration? No, it is done through the scriptures, the Word of God. Faith comes by hearing the word of God (Rom 10:17) and no man can believe in him of whom they have not heard (Rom 10:14).
I think you are on to something with connecting 6:44-45, 6:60-63, and 6:37 and 6:39. I agree that Jesus explains 37 and 39 in verses 44-45 with a quotation from Jeremiah 31 (I believe anyway). I'm not sure if you are reading closely unless I am to narrow minded on my theology I can't see around it. I really don't think it is a good practice to jump to your conclusion with Romans 10 because I think you are missing the point in the context of John. Here is why I say that:
What happened is that Jesus fed 5,000 + people then they came back for Him a day later looking for Him. Their motive was to get more food. Jesus knew this and rebuked them by saying,
"You seek Me, not because you saw signs, but because you ate of he loaves and were filled" (6:26). Jesus then went on to tell them to
"work for food which endures to eternal life." As a result they asked what this work God wanted them to do was (6:28). Jesus then showed them that they must believe in Him (6:29). They then asked for sign so they could know or believe using the example of the time where God fed the Israelites with bread from heaven. Jesus took their example and applied it to Himself (6:32-33).
So that sets the context. Jesus is definitely speaking to a group of Jews here (6:59). I am not sure if any Gentiles are present. It is a blessing that we all agree on the main and simple point of this passage that is that we must believe in Jesus. That is John's main point here in line with his purpose statement (20:30-31). So rest assured both Cals and Non Cals agree on the main point. We must, by our own choice, come to Christ, and believe in Him.
So now we come to our text at hand.
6:35- Jesus restates that He is the bread of life, the bread that God has given to save His people eternally by paying for their sins. He is unlike the bread in Moses day because that gave them physical life. Jesus brings Spiritual life. "he who comes to Me will never thirst" reveals that we must willingly come to Christ personally and trust in Him and then we will never thirst (have eternal quenching from sin). This is yet another statement teaching faith is the means to eternal life.
6:36- The Jews here are rebuked by Jesus. He said, "you see Me, and yet do not believe." Jesus has brought many signs and wonders and clear teachings yet they just simply cannot except the fact that He is their Messiah (see also 3:2, 5:31-40). Many factors have prevented them and in their case Isaiah was right as repeated the first 6 books of the New Testament (Is. 6:9-10; John 12:37-43). The Jews were blind and God had left them in their judicial punishment and was beginning to work the way for Gentiles to be brought into His kingdom (see also John 10:14-16, 25-29) first while the Jews stumbled on the rock of offense (the blindness of the Law). The Jews were blind, blind guides, dead, unable to come to Christ, and on and on we can read about the impossibility of the ones who were left in their sinful state. Yet some Jews did believe and God revealed His arm to them (refer back to my John 12 reference). So we see the immediate need for God's work.
6:37- The need of God to work in them in Jesus mind was an essential element to faith. Jesus said, "All the Father gives Me will come to Me, and the one who comes to Me I will certainly not cast out." Jesus already has told us that those who believe get life and they are free to come. He also said they refused to believe, and now He is saying that the ones who come to Him are precisely those who God has elected or chosen to be His sheep (refer back to my John 10 references). In order to come to Christ in faith it is essential for God to be the one drawing you (This is where 6:44-45 fit in). It isn't that Jesus will turn people down because they are not elect. Many people have the idea that people are clawing at Jesus' feet and He's like well let me check the list, nope, your not on the list get out. In this passage Jesus said, "The one that comes to Me I will surely not cast out." This is showing the connection between the elect and the sovereign work of God in the elect revealing that everyone who comes are the ones who God has elected for Jesus to save. Words to notice in the passage are "ALL THE FATHER GIVES WILL COME." This proves that not everyone is given. This is why many are charged with universalism and then they become hostile because they don't see this or intend to teach universalism.
6:38 and 39- Jesus came to do the will of God. In God's will ALL that Jesus is given WILL COME. Do you see election and predestination here? Everyone God gives to Jesus to save, will come, and Jesus raise them to glory on the day of judgment. Jesus said, "This is the will of Him who sent Me, that all that He has given Me I lose nothing, but raise it up on the last day." There is a strong connection between the work of the Spirit before justification, during sanctification, and in glorification. I can go outside of the text here, but for the sake of this study I will not.
6:40- Jesus goes back to the surface teaching here that a child can understand. It doesn't matter if you fully get everything Jesus said at first. Nobody understood the teaching in John 6 because they didn't understand Jesus purpose fully and He used difficult and deep theological language. Yet, a child can understand the simple offer to come to Jesus. Remember faith in Christ is the key to the text.
In vss. 63-65 this is shown again, Jesus says the words he speaks unto us are spirit and life, but some believe not, therefore no man can come to him unless it were given by the Father. If you don't believe the words (scripture) given by the Father, you cannot come to Jesus.
I hope you spend time to read my above post, if not you will never be able to discuss this rationally with me. So please read my post and my Scripture references and think about it.
I would be happy to explain how 6:63-65 fits in the picture but I want you to deal with the above posting first. Remember belief if determined by God's predetermined will. Those who come (in faith) are saved and those exact ones who come are those who God has given to the Son to save. The one's given are the ones drawn and taught by the Father through the Spirit. The Spirit gives (or brings) life to the souls of men. Through Jesus words the Spirit was working to teach the hearts of men and bring life to then in accordance with the predetermined will of God. verse 64 plainly restates that some are in unbelief why? John 10:26, "You do not believe because you are not of My sheep." God's elect sheep hear the voice of the Son and come to their Shepherd and the sheep are from Israel and now also from Gentiles (10:14-16).
So in 6:65- people don't believe and that is exactly why He said (6:44), "No one can come to Me unless the Father who sent Me draws him." belief in Christ is a result of God's election and drawing (election: 6:37 and 39; they definitely come who God has given).